# If $A=\begin{bmatrix}1&x\\x^2&4y\end{bmatrix}$ and $B=\begin{bmatrix}-3&1\\1&0\end{bmatrix}$ and adj$(A+B)=\begin{bmatrix}1 &0\\0&1\end{bmatrix}$ then values of $x$ and $y$ are

$(a)\;1,1\qquad(b)\;\pm 1,1\qquad(c)\;1,0\qquad(d)\;None\;of\;these$

$A=\begin{bmatrix}1 &x\\x^2&4y\end{bmatrix}$ and $B=\begin{bmatrix}-3&1\\1&0\end{bmatrix}$
$adj A=\begin{bmatrix}4y&-x\\-x^2&1\end{bmatrix}$
$adjB=\begin{bmatrix}-3&1\\1&0\end{bmatrix}$
$adj(A+B)=\begin{bmatrix}4y&-x\\-x^2&1\end{bmatrix}+\begin{bmatrix}-3&1\\1&0\end{bmatrix}$
$\Rightarrow \begin{bmatrix}1 &0\\0&1\end{bmatrix}=\begin{bmatrix}4y-3&-x+1\\-x^2+1&1+0\end{bmatrix}$
$\Rightarrow 4y-3=1$
$4y=1+3$
$y=\large\frac{4}{4}$$=1$
$\Rightarrow -x+1=0\Rightarrow x=1$
Hence (a) is the correct answer.
edited Mar 22, 2014